Physical Chemistry III (728342)

The Schrödinger EquationPiti TreesukolKasetsart UniversityKamphaeng Saen Campus

http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon

Solving Schrödinger Equations Eigenvalue problem

Simple cases• The free particle • The particle in a box • The finite potential well • The particle in a ring • The particle in a spherically

symmetric potential • The quantum harmonic oscillator • The hydrogen atom or hydrogen-like

atom • The particle in a one-dimensional

lattice (periodic potential)

VEmdxd

ExVdxd

m

22

2

2

22

2

)(2

Solutions for a Particle in a 1-D Box Solve for possible r and s in

that satisfy the Schrödinger equation.

Not all solutions are acceptable. The wave function needs to be continuous in all regions

sxBrxA cossin

a0for )2(cos)2(sin

)2(12/112/1

12/1

xxmEBxmEA

mEsr

II

x

0I 0III conditionsboundary 0)(0)0(

aII

II

Using the boundary conditions

• Only these allowed energy can make the wavefunction well-behaved.

Solutions for a Particle in a 1-D Box

0 00cos0sin)0( BBAII

...3,2,1 ;8

)2( 0)2(sin

0)2(sin)(

2

22

2/112/1

12/1

nma

hnE

namEamE

amEAaII

n=1n=2

n=3

n=4En

ergy

E1

E2

E3

E4

Solutions for a Particle in a 1-D Box Normalizability of the

wavefunction

The lowest energy is when n=1 For Macroscopic system,

n is very large

axa

xna

aiA

cxdxcx

dxa

xnAdxdx

a

aa

II

0for sin2

2

41

2sin

sin||||||1

2/1

2/1

0

2

0

22

0

22

Normalization constant

2

2

8mahE Zero-point

energy

groundstate

Solutions for a Particle in a 3-D Box For the 3-D box (axbxc), the

potential outside the box is infinity and is zero inside the box

Edzd

dyd

dxd

m

2

2

2

2

2

22

2

)()()( zZyYxX

EzZzZ

myYyY

mxXxX

m

)()("

22

)()("

22

)()("

22

2

222/1

2

222/1

2

222/1

8 ;sin2)(

8 ;sin2)(

8 ;sin2)(

mchnE

czn

czZ

mbhn

Eb

ynb

yY

mahnE

axn

axX

zz

z

yy

y

xx

x

Solutions for a Particle in a 3-D Box

• Inside the box:• Outside the box:

• Total energy

• The ground state is when nx=1 ny=1 nz=1

0

sinsinsin8 2/1

c

znb

yna

xnabc

zyx

2

2

2

2

2

22

8 cn

bn

an

mhE zyx

Degeneracy A Particle in a 3-D box with

a=b=c

• States have the same energy.

An energy level corresponding to more than one states is said to be degenerate.

The number of different state belonging to the level is the degree of degeneracy.

sinsinsin2 2/3

czn

byn

axn

azyx

2222

28 zyx nnnmahE

, , 112121211,, zyx nnn

Orthogonality and the Bracket Notation Two wavefunctions are

orthogonal if the integral of their product vanishes

Dirac Bracket Notation 0* dmn

mnmn

mn

nm

nm

0 1

Kronecker Delta

)( 0 mnmnmn

1 nnnn

m

n

m

n

* bra

ket

A Finite Depth Potential Box A potential well with a finite

depth Classically

• Particles with higher energy can get out of the box

• Particles with lower energy is trapped inside the box

Ener

gy

x

A Finite Depth Potential Box If the potential energy does not

rise to infinity at the wall and E < V, the wavefunction does not decay abruptly to zero at the wall• X < 0 V = 0• X > 0 V = constant• X > L V = 0

Ener

gy

x=0

Particle’s energy

Barrier Height

x=L

The Tunneling Effect

2/12 0 mEkBeAex ikxikx

2/1

2

2

2 2

0

VEmDeCe

EVdxd

m x

xixi

2/12 mEkFeEeLx ikxikx

Ener

gy

x=0 x=L

Incident wave

Reflected waveTransmitted wave

The Tunneling Effect Boundary conditions (x=0, x=L)

• F = 0 because there is no particle traveling to the left on the right of the barrier

iklikLLL FeEeDeCeLx

DCBAx

0

xbc

xbc

bcbc

ji

ji

)()(

)()(

FunctionFirst Derivative

Function

First Derivative

iklikLLL ikFeikEeDeCeLx

DCikBikAx

0

2

2

)

)

EProb(

AProb(

dtransmitte

incident

The Tunneling Effect Transmission

Probability

• For high, wide barriers , the probability is simplified to

• The transmission probability decrease exponentially with the thickness of the barrier and with m1/2.

The leakage by penetration through classically forbidden zones is called tunneling.

VE

eeA

ET

LL

incident

dtransmitte

/ when

1161

12

2

2

2

2

LeT 2116

1L

Tran

smiss

ion

Coeffi

cient

E/V00.0 1.0 2.0 3.0 4.0

1.0

0.5

0.0

The Harmonic Oscillator (Vibration) A Harmonic Oscillator and

Hooke’s Law• Harmonic motion: Force is

proportional to its displacement: F= –kx when k is the force constant.

l0

l

l0

)(

)(

2

2

02

2

0

kxdt

xdm

llkdt

ldm

kxllkf

21

cossin)( 21

mk

tctctx

2

21)(

)(

kxxV

dxdVxf

Harmonic Potential

x

V

0Equilibrium

The 1-D Harmonic Oscillator Symmetrical well

Schrödinger equation with harmonic potential:

Ekxdxd

m

kxxV

22

22

2

21

2

21)(

Solving the equation by using boundary conditions that .

The permitted energy levels are

The zero-point energy of a harmonic oscillator is

... 3 2, 1, ,0 2/1

21

v

mkvEv

0

21

0 EDisplacement,

x

Pote

ntia

l En

ergy

, V

0

1

2

3

4

5

6

7

2)( axxV

Solutions for 1-D Harmonic Oscillator Wavefunctions for a harmonic

oscillator

• Hermite polynomials;

4/12

2/)/( 2

)(

.)(

mkexHNx

fnGaussianxinpolynomialNx

x

12072048064)(

12016032)(

124816)(

128

24

21

2466

355

244

33

22

1

0

yyyyH

yyyyH

yyyH

yyyH

yyH

yyHyH

yH

2/)/(00

2

)( xeNx

2/)/(11

22)(

xexNx

2/)/(2

2

22

2

24)(

xexNx

2/)/(3

3

33

2

128)(

xexxNx

Wavefunctions

2/1

2/1 !21

N

Normalizing Factors 2/1

mk

0

1

2

3

At high quantum number (>>) harmonic oscillator has their highest amplitudes near the turning points of the classical motion (V=E)The properties of

oscillators• Observables

• Mean displacement 0

1

2

3

4

5

6

7

n

ˆˆ* dx

0

)()(

)()(

2

2

22

2

2

2

2

121

122

22

2/2/22

222*

dyeHHHN

dyeyHHN

dyeHyeHN

dxeHxeHNdxxx

y

y

yy

xx

' if !2' if 0

2/1'

2

dyeHH y

8%

• Mean square displacement

• Mean potential energy

• Mean kinetic energy

The tunneling probability decreases quickly with increasing .

Macroscopic oscillators are in states with very high quantum number.

2/1212

)(mkx

Emk

kkxV

21

21

21

2/121

212

21

)(

EVEEK 21

Rigid Rotor (Rotation) 2-D Rotation

• A particle of mass m constrained to move in a circular path of radius r in the xy plane.E = EK+VV = 0EK = p2/2m

• Angular momentum Jz= pr• Moment of inertia I = mr2

x

y

z

r

IJE

mrIprJm

pE

z

z

2

;;2

2

22

Not all the values of the angular momentum are permitted!

• Using de Broglie relation, the angular momentum about the z-axis is

• A particle is restricted to the circular path thus cannot take arbitrary value, otherwise it would violate the requirements for satisfied wavefunction.

hrJ

hpprJ z

z

;

lmr 2

Allowed wavelengths

• The angular momentum is limited to the values

• The possible energy levels are

,2,1,0 22

l

lll

z

m

mhmr

hrmhrJ

ml > 0

ml < 0

Im

IJE lz

22

222

Solutions for 2-D rotation Hamiltonian of 2-D rotation

• The radius of the path is fixed then

The Schrödinger equation is

2

2

22

22

2

2

2

22

112

,

2,

rrrrmrH

yxmyxH

2

22

2

2

2

2

22 dd

Idd

mrH

22

2 2IE

dd

2/1

2/1

2)2(

)( IEmel

im

m

l

l

• Cyclic boundary condition

• The probability density is independent of

l

l

lll

l

mim

imimim

m

e

eee

2

2/1

2

2/1

2

)(

)2()2()2(

)2()(

1ie

,2,1,0

1

)(1)2(2

2

l

m

mm

m

m

l

l

l

l

positive be must

21

22 2/12/1*

ll

ll

imim

mmee

Spherical Coordinates Coordinates

defined by r, , *

www.mathworld.wolfram.com

0 20

0

angle Polar

angle Azimuthal

Radius r

rzxy

zyxr

1

1

222

cos

tan

cossinsinsincos

rzryrx

θ̂sin1φ̂1r̂

sinV

r̂sina

θ̂sinφ̂r̂s

2

2

rrr

drddrd

drd

drrddrd

sinsin

1sin

1122

2

222

22

rrrr

3-D Rotation A particle of mass m that

free to move anywhere on the surface of a sphere radius r.• The Schrödinger equation

rV

mH 2

2

2

fixed is

travel to free is it whenever

0r

V

)()(),(),,(2

22

r

Em

Using spherical coordinate

• Discard terms that involve differentiation wrt. r

sinsin

1sin

1

11

2

2

22

222

22

rrrr

sinsin

1sin

1112

2

222

22

rr

2

22

22

;2

2

21

mrIIE

mEr

mEr

Legendrian

Plug the separable wavefunction into the Schrödinger equation

The equation can be separated into two equations

22

2

2

2

2

2

2

2

2

sinsinsin1

sinsinsin

sinsin

1sin

1

dd

dd

dd

dd

dd

dd

222

2

sinsinsin1

dd

ddm

dd

l

The normalized wavefunctions are denoted , which depend on two quantum numbers, l and ml, and are called the spherical harmonics.

Solutions for 3-D Rotation),(,

lmlY

0 0

10

1

2

0

1

2

),(, lmlYl lm

2/1

41

cos43 2/1

ie

sin

83 2/1

1cos316

5 22/1

ie

sincos

85 2/1

ie 222/1

sin32

5

llllml l ,,2,1,2,1,0

• For a given l, the most probable location of the particle migrates towards the xy-plane as the value of |ml| increases

• The energy of the particle is restricted to the values

Energy is quantized.Energy is independent of ml values.A level with quantum number l is (2l+1) degenerate.

,2,1,02

)1( lI

llE

Angular Momentum & Space Quantization Magnitude of angular

momentum z-component of angular

momentum The orientation of a rotating

body is quantizedz

ml = 0

ml = +1

ml = -1

ml = +2

ml = -2

... 2, 1, ,01 2/1 lll

lllmm ll ,...,1 ,

+2

+1

0

–1

–2

Spin The intrinsic angular

momentum is called “spin”• Spin quantum number; s = ½

• Spin magnetic quantum number; ms = s, s–1, … –s

Element particles may have different s values• half-integral spin: fermions (electron, proton)

• integral spin: boson (photon)

ms = +½

ms = –½

Key Ideas Wavefunction

• Acceptable• Corresponding to boundary conditions

Modes of motion (Functions to explain the motion)• Translation Particle in a box• Vibration Harmonic oscillator• Rotation Rigid rotor

Tunneling effect